Heat 1 1/2 teaspoons of the oil over medium-high heat in a large Dutch oven. Add the turkey and cook, breaking up chunks with the side of a wooden spoon, until browned, about 5 minutes. Push the turkey to the edges of the pan, leaving the middle empty. Reduce the heat to medium and add the remaining 1 1/2 teaspoons oil, then the onion and garlic to the center of the pan. Cook, stirring occasionally, until the vegetables start to soften, about 3 minutes. Add the chili powder, cumin and coriander and stir about 30 seconds. Add the tomato paste and 1 teaspoon salt and stir until the paste begins to darken in color, about 30 seconds. Then add the squash, tomatoes and 4 cups water, scraping the bottom of the pan to release any stuck bits. Bring to a simmer, adjust the heat and simmer, uncovered, until the chili has thickened and the squash is tender, 35 to 40 minutes.
Stir the beans and chia seeds into the chili and heat through, about 5 minutes. Season with an additional 1/2 teaspoon salt and a few grinds of pepper, then taste the chili and stir in up to 2 tablespoons of vinegar. Spoon the chili into bowls and top each bowl with cilantro and a dollop of sour cream.
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